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2006 Cyprus MO/Lyceum/Problems

Revision as of 12:15, 5 July 2007 by Leonariso (talk | contribs) (→‎Problem 21)

Problem 1

A diary industry, in a quantity of milk with $4%$ (Error compiling LaTeX. Unknown error_msg) fat adds a quantity of milk with $1%$ (Error compiling LaTeX. Unknown error_msg) fat and produces $1200$kg of milk with $2%$ (Error compiling LaTeX. Unknown error_msg) fat. The quantity of milk with $1%$ (Error compiling LaTeX. Unknown error_msg) fat, that was added is (in kg)

A. $1000$

B. $600$

C. $800$

D. $120$

E. $480$

Solution

Problem 2

The operation $\alpha * \beta$ is defined by $\alpha * \beta = \alpha^2 - \beta^2$ $\forall \alpha , \beta \in R$. The value of the expression $K = \left[\left(1+\sqrt{3}\right) * 2\right]*\sqrt{2}$ is

A. $3$

B. $0$

C. $\sqrt{3}$

D. $9$

E. $1$

Solution

Problem 3

The domain of the function $f(x)=\sqrt{4+2x}$ is

A. $(-2,+\infty)$

B. $[0,+\infty)$

C. $[-2,+\infty)$

D. $[-2,0]$

E. $R$

Solution

Problem 4

Given the function $f(x)=\alpha x^2 +9x+ \frac{81}{4\alpha}$ , $\alpha \neq 0$ Which of the following is correct, about the graph of $f$?

A. intersects x-axis

B. touches y-axis

C. touches x-axis

D. has minimum point

E. has maximum point

Solution

Problem 5

If both integers $\alpha,\beta$ are bigger than 1 and satisfy $a^7=b^8$, then the minimum value of $\alpha+\beta$ is

A. $384$

B. $2$

C. $15$

D. $56$

E. $512$

Solution

Problem 6

The value of the expression $K=\sqrt{19+7\sqrt{3}}-\sqrt{7+4\sqrt{3}}$ is

A. $4$

B. $4\sqrt{3}$

C. $12+4\sqrt{3}$

D. $-2$

E. $2$

Solution

Problem 7

2006 CyMO-7.PNG

In the figure, $ABC$ is an equilateral triangle and $AD\perp BC$, $DE\perp AC$, $EZ\perp BC$. If $EZ=\sqrt{3}$, then the length of the side os the triangle ABC is

A. $\frac{3\sqrt{3}}{2}$

B. $8$

C. $4$

D. $3$

E. $9$

Solution

Problem 8

2006 CyMO-8.PNG

In the figure $ABCDE$ is a regular 5-sided polygon and $Z$, $H$, $L$, $I$, $K$ are the points of intersections of the extensions of the sides. If the area of the "star" $AHCLCIDKEZA$ is 1, then the area of the shaded quadrilateral $ACIZ$ is

A. $\frac{2}{3}$

B. $\frac{1}{2}$

C. $\frac{3}{7}$

D. $\frac{3}{10}$

E. None of these

Solution

Problem 9

If $x=\sqrt[3]{4}$ and $y=\sqrt[3]{6}-\sqrt[3]{3}$, then which of the following is correct

A. $x=y$

B. $x<y$

C. $x=2y$

D. $x>2y$

E. None of these

Solution

Problem 10

If $2^x=15$ and $15^y=256$, then the product $xy$ equals

A. $7$

B. $3$

C. $1$

D. $8$

E. $6$

Solution

Problem 11

2006 CyMO-11.PNG

The lines $(\epsilon):x-2y=0$ and $(\delta):x+y=4$ intersect at the point $C$. If the line $(\delta)$ intersects the axes $Ox$ and $Oy$ to the points $A$ and $B$ respectively, then the ratio of the area of the triangle $OAC$ to the area of the triangle $OBC$ equals

A. $\frac{1}{3}$

B. $\frac{2}{3}$

C. $\frac{3}{5}$

D. $\frac{1}{2}$

E. $\frac{4}{9}$

Solution

Problem 12

If $f(\alpha,\beta)= \begin{cases} \displaystyle \alpha & \textrm {if} \qquad \alpha=\beta \\ f(\alpha-\beta,\beta) & \textrm {if} \qquad \alpha>\beta \\ f(\beta-\alpha,\alpha) & \textrm {if} \qquad \alpha<\beta \end{cases}$

then $f(28,17)$ equals

A. $8$

B. $0$

C. $11$

D. $5$

E. $1$

Solution

Problem 13

The sum of the digits of the number $10^{2006}-2006$ is

A. $18006$

B. $20060$

C. $2006$

D. $18047$

E. None of these

Solution

Problem 14

2006 CyMO-17.PNG

The rectangle $ABCD$ is a small garden divided to the rectangle $AXED$ and to the square $ZBCE$, so that $AE=2\sqrt{5}m$ and the shaded area of the triangle $DBE$ is $4m^2$. The area of the whole garden is

A. $24m^2$

B. $20m^2$

C. $16m^2$

D. $32m^2$

E. $10\sqrt5m^2$

Solution

Problem 15

The expression :$\frac{1}{2+\sqrt7} + \frac{1}{\sqrt7+\sqrt{10}}+ \frac{1}{\sqrt{10}+\sqrt{13}} + \frac{1}{\sqrt{13}+4}$ equals

A. $3/4$

B. $\frac{3}{2}$

C. $\frac{2}{5}$

D. $\frac{1}{2}$

E. $\frac{2}{3}$

Solution

Problem 16

If $x_1,x_2$ are the roots of the equation $x^2-2kx+2m=0$, then $\frac{1}{x_1},\frac{1}{x_2}$ are the roots of the equation

A. $x^2-2k^2x+2m^2=0$

B. $x^2-\frac{k}{m}x+\frac{1}{2m}=0$

C. $x^2-\frac{m}{k}x+\frac{1}{2m}=0$

D. $2mx^2-kx+1=0$

E. $2kx^2-2mx+1=0$

Solution

Problem 17

2006 CyMO-17.PNG

$ABC$ is equilateral triangle of side $\alpha$ and $AD=BE=\frac{\alpha}{3}$. The measure of the angle $\ang CPE$ (Error compiling LaTeX. Unknown error_msg) is

A. $60^{\circ}$

B. $50^{\circ}$

C. $40^{\circ}$

D. $45^{\circ}$

E. $70^{\circ}$

Solution

Problem 18

2006 CyMO-18.PNG

$K(k,0)$ is the minimum point of the parabola and the parabola intersects the y-axis at the point $C(0,k)$. If the area if the rectangle $OABC$ is $8$, then the equation of the parabola is

A. $y=\frac{1}{2}(x+2)^2$

B. $y=\frac{1}{2}(x-2)^2$

C. $y=x^2+2$

D. $y=x^2-2x+1$

E. $y=x^2-4x+4$

Solution

Problem 19

2006 CyMO-19.PNG


Solution

Problem 20

Solution

Problem 21

A convex polygon has $n$ sides and $740$ diagonals. Then $n$ equals

A. $30$

B. $40$

C. $50$

D. $60$

E. None of these

Solution

Problem 22

2006 CyMO-22.PNG


Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

Problem 26

Solution

Problem 27

Solution

Problem 28

Solution

Problem 29

Solution

Problem 30

Solution

See also

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